WireGuard uses keys that are on the Montgomery curve, not the Edwards curve. It's the standard X25519 or Curve25519 commonly used for ECDH.
crypto_box_curve25519xsalsa20poly1305_keypair which calls
randombytes_buf(sk, 32), return crypto_scalarmult_curve25519_base(pk, sk). In WireGuard,
wg genkey calls
get_random_bytes(sk, 32) and
wg pubkey calls
curve25519_generate_public(pk, sk). It turns out that
crypto_scalarmult_curve25519_base(pk, sk) and
curve25519_generate_public(pk, sk) are essentially the same thing.
So this means in theory those keys would be "compatible".
However, you should not do this under any circumstances.
Cryptographic protocols make various assumptions on how keys are used. The WireGuard Protocol is formally verified, which means within the WireGuard handshake, it's been verified that none of the constructions will lead to any catastrophic situations. However, there's no guarantee that re-using these same keys within other random constructions that you might cook up won't interact badly and will retain the same guarantees of the WireGuard Protocol. So, if you reuse WireGuard private keys in your own protocol, you do so at your own risk. That's the kind of thing you only do if you really, really know what you're doing and are certain that there won't be any bad interactions.
(Perhaps a more productive way forward would be to describe what you actually want to do with WireGuard key identities and your libsodium-based protocol. Then maybe folks can help you find a safe way of accomplishing this. What's the overall end goal of your project?)